# Solved game

Last updated

A solved game is a game whose outcome (win, lose or draw) can be correctly predicted from any position, assuming that both players play perfectly. This concept is usually applied to abstract strategy games, and especially to games with full information and no element of chance; solving such a game may use combinatorial game theory and/or computer assistance.

## Overview

A two-player game can be solved on several levels: [1] [2]

Ultra-weak
Prove whether the first player will win, lose or draw from the initial position, given perfect play on both sides. This can be a non-constructive proof (possibly involving a strategy-stealing argument) that need not actually determine any moves of the perfect play.
Weak
Provide an algorithm that secures a win for one player, or a draw for either, against any possible moves by the opponent, from the beginning of the game. That is, produce at least one complete ideal game (all moves start to end) with proof that each move is optimal for the player making it. It does not necessarily mean a computer program using the solution will play optimally against an imperfect opponent.
Strong
Provide an algorithm that can produce perfect moves from any position, even if mistakes have already been made on one or both sides.

Despite their name, many game theorists believe that "ultra-weak" proofs are the deepest, most interesting and valuable. "Ultra-weak" proofs require a scholar to reason about the abstract properties of the game, and show how these properties lead to certain outcomes if perfect play is realized.[ citation needed ]

By contrast, "strong" proofs often proceed by brute force—using a computer to exhaustively search a game tree to figure out what would happen if perfect play were realized. The resulting proof gives an optimal strategy for every possible position on the board. However, these proofs are not as helpful in understanding deeper reasons why some games are solvable as a draw, and other, seemingly very similar games are solvable as a win.

Given the rules of any two-person game with a finite number of positions, one can always trivially construct a minimax algorithm that would exhaustively traverse the game tree. However, since for many non-trivial games such an algorithm would require an infeasible amount of time to generate a move in a given position, a game is not considered to be solved weakly or strongly unless the algorithm can be run by existing hardware in a reasonable time. Many algorithms rely on a huge pre-generated database, and are effectively nothing more.

As an example of a strong solution, the game of tic-tac-toe is solvable as a draw for both players with perfect play (a result even manually determinable by schoolchildren). Games like nim also admit a rigorous analysis using combinatorial game theory.

Whether a game is solved is not necessarily the same as whether it remains interesting for humans to play. Even a strongly solved game can still be interesting if its solution is too complex to be memorized; conversely, a weakly solved game may lose its attraction if the winning strategy is simple enough to remember (e.g. Maharajah and the Sepoys). An ultra-weak solution (e.g. Chomp or Hex on a sufficiently large board) generally does not affect playability.

Moreover, even if the game is not solved, it is possible that an algorithm yields a good approximate solution: for instance, an article in Science from January 2015 claims that their heads up limit Texas hold 'em poker bot Cepheus guarantees that a human lifetime of play is not sufficient to establish with statistical significance that its strategy is not an exact solution. [3] [4] [5]

## Perfect play

In game theory, perfect play is the behavior or strategy of a player that leads to the best possible outcome for that player regardless of the response by the opponent. Perfect play for a game is known when the game is solved. [1] Based on the rules of a game, every possible final position can be evaluated (as a win, loss or draw). By backward reasoning, one can recursively evaluate a non-final position as identical to the position that is one move away and best valued for the player whose move it is. Thus a transition between positions can never result in a better evaluation for the moving player, and a perfect move in a position would be a transition between positions that are equally evaluated. As an example, a perfect player in a drawn position would always get a draw or win, never a loss. If there are multiple options with the same outcome, perfect play is sometimes considered the fastest method leading to a good result, or the slowest method leading to a bad result.

Perfect play can be generalized to non-perfect information games, as the strategy that would guarantee the highest minimal expected outcome regardless of the strategy of the opponent. As an example, the perfect strategy for rock paper scissors would be to randomly choose each of the options with equal (1/3) probability. The disadvantage in this example is that this strategy will never exploit non-optimal strategies of the opponent, so the expected outcome of this strategy versus any strategy will always be equal to the minimal expected outcome.

Although the optimal strategy of a game may not (yet) be known, a game-playing computer might still benefit from solutions of the game from certain endgame positions (in the form of endgame tablebases), which will allow it to play perfectly after some point in the game. Computer chess programs are well known for doing this.

## Solved games

Awari (a game of the Mancala family)
The variant of Oware allowing game ending "grand slams" was strongly solved by Henri Bal and John Romein at the Vrije Universiteit in Amsterdam, Netherlands (2002). Either player can force the game into a draw.
Chopsticks
The second player can always force a win.[ citation needed ]
Connect Four
Solved first by James D. Allen on October 1, 1988 and independently by Victor Allis on October 16, 1988. [6] The first player can force a win. Strongly solved by John Tromp's 8-ply database [7] (Feb 4, 1995). Weakly solved for all boardsizes where width+height is at most 15 (as well as 8×8 in late 2015) [6] (Feb 18, 2006).
English draughts (checkers)
This 8×8 variant of draughts was weakly solved on April 29, 2007 by the team of Jonathan Schaeffer. From the standard starting position, both players can guarantee a draw with perfect play. [8] Checkers is the largest game that has been solved to date, with a search space of 5×1020. [9] The number of calculations involved was 1014, which were done over a period of 18 years. The process involved from 200 desktop computers at its peak down to around 50. [10]
Fanorona
Weakly solved by Maarten Schadd. The game is a draw.[ citation needed ]
Free gomoku
Solved by Victor Allis (1993). The first player can force a win without opening rules.
Ghost
Solved by Alan Frank using the Official Scrabble Players Dictionary in 1987.[ citation needed ]
Guess Who?
Strongly solved by Mihai Nica in 2016. [11] The first player has a 63% chance of winning under optimal play by both sides.
Hex
• A strategy-stealing argument (as used by John Nash) shows that all square board sizes cannot be lost by the first player. Combined with a proof of the impossibility of a draw this shows that the game is ultra-weak solved as a first player win.
• Strongly solved by several computers for board sizes up to 6×6.
• Jing Yang has demonstrated a winning strategy (weak solution) for board sizes 7×7, 8×8 and 9×9.
• A winning strategy for Hex with swapping is known for the 7×7 board.
• Strongly solving Hex on an N×N board is unlikely as the problem has been shown to be PSPACE-complete.
• If Hex is played on an N×(N+1) board then the player who has the shorter distance to connect can always win by a simple pairing strategy, even with the disadvantage of playing second.
• A weak solution is known for all opening moves on the 8×8 board. [12]
Hexapawn
3×3 variant solved as a win for black, several other larger variants also solved. [13]
Kalah
Most variants solved by Geoffrey Irving, Jeroen Donkers and Jos Uiterwijk (2000) except Kalah (6/6). The (6/6) variant was solved by Anders Carstensen (2011). Strong first-player advantage was proven in most cases. [14] [15] Mark Rawlings, of Gaithersburg, MD, has quantified the magnitude of the first player win in the (6/6) variant (2015). After creation of 39 GB of endgame databases, searches totaling 106 days of CPU time and over 55 trillion nodes, it was proven that, with perfect play, the first player wins by 2. Note that all these results refer to the Empty-pit Capture variant and therefore are of very limited interest for the standard game. Analysis of the standard rule game has now been posted for Kalah(6,4), which is a win by 8 for the first player, and Kalah(6,5), which is a win by 10 for the first player. Analysis of Kalah(6,6) with the standard rules is on-going, however, it has been proven that it is a win by at least 4 for the first player.
L game
Easily solvable. Either player can force the game into a draw.
Losing chess
Weakly solved as a win for white beginning with 1. e3. [16]
Maharajah and the Sepoys
This asymmetrical game is a win for the sepoys player with correct play.
Nim
Strongly solved.
Nine men's morris
Solved by Ralph Gasser (1993). Either player can force the game into a draw. [17]
Order and Chaos
Order (First player) wins. [18]
Ohvalhu
Weakly solved by humans, but proven by computers. (Dakon is, however, not identical to Ohvalhu, the game which actually had been observed by de Voogt)
Pangki
Strongly solved by Jason Doucette (2001). [19] The game is a draw. There are only two unique first moves if you discard mirrored positions. One forces the draw, and the other gives the opponent a forced win in 15.
Pentago
Strongly solved. [20] The first player wins.
Pentominoes
Weakly solved by H. K. Orman. [21] It is a win for the first player.
Poddavki ("Russian Give-away Checkers")
Solved by Osipov and Morozev in 2011. A white win.[ citation needed ]
Quarto
Solved by Luc Goossens (1998). Two perfect players will always draw.
Qubic
Weakly solved by Oren Patashnik (1980) and Victor Allis. The first player wins.
Renju -like game without opening rules involved
Claimed to be solved by János Wagner and István Virág (2001). A first-player win.
Sim
Weakly solved: win for the second player.
Teeko
Solved by Guy Steele (1998). Depending on the variant either a first-player win or a draw. [22]
Three men's morris
Trivially solvable. Either player can force the game into a draw.
Three Musketeers
Strongly solved by Johannes Laire in 2009, and weakly solved by Ali Elabridi in 2017. [23] It is a win for the blue pieces (Cardinal Richelieu's men, or, the enemy). [24]
Tic-tac-toe
Trivially strongly solvable because of the small game tree. [25] The game is a draw if no mistakes are made, with no mistake possible on the opening move.
Tigers and Goats
Weakly solved by Yew Jin Lim (2007). The game is a draw. [26]

## Partially solved games

Chess
Fully solving chess remains elusive, and it is speculated that the complexity of the game may preclude its ever being solved. Through retrograde computer analysis, endgame tablebases (strong solutions) have been found for all three- to seven-piece endgames, counting the two kings as pieces.
Some variants of chess on a smaller board with reduced numbers of pieces have been solved. Some other popular variants have also been solved; for example a weak solution to Maharajah and the Sepoys is an easily memorable series of moves that guarantees victory to the "sepoys" player.
Go
The 5×5 board was weakly solved for all opening moves in 2002. [27] The 7×7 board was weakly solved in 2015. [28] Humans usually play on a 19×19 board which is over 145 orders of magnitude more complex than 7×7. [29]
International draughts
All endgame positions with two through seven pieces were solved, as well as positions with 4×4 and 5×3 pieces where each side had one king or fewer, positions with five men versus four men, positions with five men versus three men and one king, and positions with four men and one king versus four men. The endgame positions were solved in 2007 by Ed Gilbert of the United States. Computer analysis showed that it was highly likely to end in a draw if both players played perfectly. [30] [ better source needed ]
m,n,k-game
It is trivial to show that the second player can never win; see strategy-stealing argument. Almost all cases have been solved weakly for k ≤ 4. Some results are known for k = 5. The games are drawn for k ≥ 8.
Reversi (Othello)
Weakly solved on a 4×4 and 6×6 board as a second player win in July 1993 by Joel Feinstein. [31] On an 8×8 board (the standard one) it is mathematically unsolved, though computer analysis shows a likely draw. No strongly supposed estimates other than increased chances for the starting player (Black) on 10×10 and greater boards exist.

## Related Research Articles

Hex is a strategy board game for two players played on a hexagonal grid, theoretically of any size and several possible shapes, but traditionally as an 11×11 rhombus. Players alternate placing markers or stones on unoccupied spaces in an attempt to link their opposite sides of the board in an unbroken chain. One player must win; there are no draws. The game has deep strategy, sharp tactics and a profound mathematical underpinning related to the Brouwer fixed-point theorem. It was invented in the 1940s independently by two mathematicians, Piet Hein and John Nash. The game was first marketed as a board game in Denmark under the name Con-tac-tix, and Parker Brothers marketed a version of it in 1952 called Hex; they are no longer in production. Hex can also be played with paper and pencil on hexagonally ruled graph paper.

Draughts or checkers is a group of strategy board games for two players which involve diagonal moves of uniform game pieces and mandatory captures by jumping over opponent pieces. Draughts developed from alquerque. The name derives from the verb to draw or to move.

Computer chess includes both hardware and software capable of playing chess. Computer chess provides opportunities for players to practice even in the absence of human opponents, and also provides opportunities for analysis, entertainment and training.

Combinatorial game theory (CGT) is a branch of mathematics and theoretical computer science that typically studies sequential games with perfect information. Study has been largely confined to two-player games that have a position in which the players take turns changing in defined ways or moves to achieve a defined winning condition. CGT has not traditionally studied games of chance or those that use imperfect or incomplete information, favoring games that offer perfect information in which the state of the game and the set of available moves is always known by both players. However, as mathematical techniques advance, the types of game that can be mathematically analyzed expands, thus the boundaries of the field are ever changing. Scholars will generally define what they mean by a "game" at the beginning of a paper, and these definitions often vary as they are specific to the game being analyzed and are not meant to represent the entire scope of the field.

The Shannon switching game is a connection game for two players, invented by American mathematician and electrical engineer Claude Shannon, the "father of information theory" some time before 1951. Two players take turns coloring the edges of an arbitrary graph. One player has the goal of connecting two distinguished vertices by a path of edges of their color. The other player aims to prevent this by using their color instead. The game is commonly played on a rectangular grid; this special case of the game was independently invented by American mathematician David Gale in the late 1950s and is known as Gale or Bridg-It.

Connect Four is a two-player connection board game, in which the players choose a color and then take turns dropping colored discs into a seven-column, six-row vertically suspended grid. The pieces fall straight down, occupying the lowest available space within the column. The objective of the game is to be the first to form a horizontal, vertical, or diagonal line of four of one's own discs. Connect Four is a solved game. The first player can always win by playing the right moves.

Combinatorial game theory has several ways of measuring game complexity. This article describes five of them: state-space complexity, game tree size, decision complexity, game-tree complexity, and computational complexity.

An m,n,k-game is an abstract board game in which two players take turns in placing a stone of their color on an m×n board, the winner being the player who first gets k stones of their own color in a row, horizontally, vertically, or diagonally. Thus, tic-tac-toe is the 3,3,3-game and free-style gomoku is the 15,15,5-game. An m,n,k-game is also called a k-in-a-row game on an m×n board.

Jonathan Herbert Schaeffer is a Canadian researcher and professor at the University of Alberta and the former Canada Research Chair in Artificial Intelligence.

Maven is an artificial intelligence Scrabble player, created by Brian Sheppard. It has been used in official licensed Hasbro Scrabble games.

Kalah, also called Kalaha or Mancala, is a game in the mancala family imported in the United States by William Julius Champion, Jr. in 1940. This game is sometimes also called "Kalahari", possibly by false etymology from the Kalahari desert in Namibia.

The Game of the Amazons is a two-player abstract strategy game invented in 1988 by Walter Zamkauskas of Argentina. It is a member of the territorial game family, a distant relative of Go and chess. El Juego de las Amazonas is a trademark of Ediciones de Mente.

English draughts or checkers, also called American checkers or straight checkers, is a form of the strategy board game draughts. It is played on an 8×8 chequered board with 12 pieces per side. The pieces move and capture diagonally forward, until they reach the opposite end of the board, when they are crowned and can thereafter move and capture both backward and forward.

An endgame tablebase is a computerized database that contains precalculated exhaustive analysis of chess endgame positions. It is typically used by a computer chess engine during play, or by a human or computer that is retrospectively analysing a game that has already been played.

God's algorithm is a notion originating in discussions of ways to solve the Rubik's Cube puzzle, but which can also be applied to other combinatorial puzzles and mathematical games. It refers to any algorithm which produces a solution having the fewest possible moves, the idea being that only an omniscient being would know an optimal step from any given configuration.

Zillions of Games is a commercial general game playing system developed by Jeff Mallett and Mark Lefler in 1998. The game rules are specified with S-expressions, Zillions rule language. It was designed to handle mostly abstract strategy board games or puzzles. After parsing the rules of the game, the system's artificial intelligence can automatically play one or more players. It treats puzzles as solitaire games and its AI can be used to solve them.

In game theory, a two-player deterministic perfect information turn-based game is a first-player-win if with perfect play the first player to move can always force a win. Similarly, a game is second-player-win if with perfect play the second player to move can always force a win. With perfect play, if neither side can force a win, the game is a draw.

Solving chess means finding an optimal strategy for playing chess, i.e. one by which one of the players can always force a victory, or both can force a draw. It also means more generally solving chess-like games, such as infinite chess. According to Zermelo's theorem, a hypothetically determinable optimal strategy does exist for chess and chess-like games.

The following outline is provided as an overview of and topical guide to chess:

Ultimate tic-tac-toe is a board game composed of nine tic-tac-toe boards arranged in a 3 × 3 grid. Players take turns playing in the smaller tic-tac-toe boards until one of them wins in the larger tic-tac-toe board. Compared to traditional tic-tac-toe, strategy in this game is conceptually more difficult and has proven more challenging for computers. It was shown in 2020 that the game admits a winning strategy for the first player to move, meaning that the first player to move can always win assuming perfect play.

## References

1. Victor Allis (1994). "PhD thesis: Searching for Solutions in Games and Artificial Intelligence" (PDF). Department of Computer Science. University of Limburg . Retrieved 2012-07-14.
2. H. Jaap van den Herik, Jos W.H.M. Uiterwijk, Jack van Rijswijck, Games solved: Now and in the future , Artificial Intelligence 134 (2002) 277–311.
3. Bowling, M.; Burch, N.; Johanson, M.; Tammelin, O. (Jan 2015). "Heads-up limit hold'em poker is solved" (PDF). Science. 347 (6218): 145–9. CiteSeerX  . doi:10.1126/science.1259433. PMID   25574016. S2CID   3796371.
4. Philip Ball (2015-01-08). "Game Theorists Crack Poker". Nature. Nature. doi:10.1038/nature.2015.16683. S2CID   155710390 . Retrieved 2015-01-13.
5. Robert Lee Hotz (2015-01-08). "Computer Conquers Texas Hold 'Em, Researchers Say". Wall Street Journal.
6. "John's Connect Four Playground". tromp.github.io.
7. "UCI Machine Learning Repository: Connect-4 Data Set". archive.ics.uci.edu.
8. Schaeffer, Jonathan (2007-07-19). "Checkers Is Solved". Science. 317 (5844): 1518–22. doi:10.1126/science.1144079. PMID   17641166. S2CID   10274228 . Retrieved 2007-07-20.
9. "Project - Chinook - World Man-Machine Checkers Champion" . Retrieved 2007-07-19.
10. Mullins, Justin (2007-07-19). "Checkers 'solved' after years of number crunching". NewScientist.com news service. Retrieved 2007-07-20.
11. Optimal Strategy in "Guess Who?": Beyond Binary Search by Mihai Nica.
12. P. Henderson, B. Arneson, and R. Hayward[webdocs.cs.ualberta.ca/~hayward/papers/solve8.pdf, Solving 8×8 Hex ], Proc. IJCAI-09 505-510 (2009) Retrieved 29 June 2010.
13. Price, Robert. "Hexapawn". www.chessvariants.com.
14. Solving Kalah by Geoffrey Irving, Jeroen Donkers and Jos Uiterwijk.
15. Solving (6,6)-Kalaha by Anders Carstensen.
16. Watkins, Mark. "Losing Chess: 1. e3 wins for White" (PDF). Retrieved 17 January 2017.
17. Nine Men's Morris is a Draw by Ralph Gasser
18. Pangki is strongly solved as a Draw by Jason Doucette
19. Geoffrey Irving: "Pentago is a first player win" http://perfect-pentago.net/details.html
20. Hilarie K. Orman: Pentominoes: A First Player Win in Games of no chance , MSRI Publications Volume 29, 1996, pages 339-344. Online: pdf.
21. Teeko, by E. Weisstein
22. Elabridi, Ali. "Weakly Solving the Three Musketeers Game Using Artificial Intelligence and Game Theory" (PDF).
23. Three Musketeers, by J. Lemaire
24. Tic-Tac-Toe, by R. Munroe
25. Yew Jin Lim. On Forward Pruning in Game-Tree Search. Ph.D. Thesis, National University of Singapore, 2007.
26. 5×5 Go is solved by Erik van der Werf
27. "首期喆理围棋沙龙举行 李喆7路盘最优解具有里程碑意义_下棋想赢怕输_新浪博客". blog.sina.com.cn. (which says the 7x7 solution is only weakly solved and it's still under research, 1. the correct komi is 9 (4.5 stone); 2. there are multiple optimal trees - the first 3 moves are unique - but within the first 7 moves there are 5 optimal trees; 3. There are many ways to play that don't affect the result)
28. Counting legal positions in Go Archived 2007-09-30 at the Wayback Machine , Tromp and Farnebäck, accessed 2007-08-24.
29. Some of the nine-piece endgame tablebase by Ed Gilbert
30. "6×6 Othello weakly solved". Archived from the original on 2009-11-01.
• Allis, Beating the World Champion? The state-of-the-art in computer game playing. in New Approaches to Board Games Research.